Add pre-rendered fire to scene
You can pre-render only the particle effects, then add a frame of the animation using Add Images as Planes (enable the add-on).

Then parent the image plane to the backdrop and change the source from a still image to a video sequence.

You can separate them out into different layers and then composite them together in the Compositor.

If you render with alpha, use the Alpha Over node. If you render against a black background use a MixRGB node set to "add".

The Compositor will also allow you to add things like glow effects if you want by using a Blur node.

Probably worth testing your workflow on a small number of frames first to make sure everything's working before rendering the whole thing.